What cosine function represents an amplitude of 3, a period of π, no horizontal shift, and a vertical shift of?

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What cosine function represents an amplitude of 3, a period of π, no horizontal shift, and a vertical shift of?

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Answer 1 · 2017-09-12T12:20:39

In order to answer this I have assumed a vertical shift of $+ 7$ $+7$

$3 cos (2 θ) + 7$ $color(red)(3cos(2theta)+7)$

Explanation:

The standard cos function $cos (γ)$ $color(green)(cos(gamma))$ has a period of $2 π$ $2pi$

If we want a period of $π$ $pi$ we need to replace $γ$ $gamma$ with something that will cover the domain "twice as fast" e.g. $2 θ$ $2theta$ .
That is $cos (2 θ)$ $color(magenta)(cos(2theta))$ will have a period of $π$ $pi$ .

To get an amplitude of $3$ $3$ we need to multiply all values in the Range generated by $cos (2 θ)$ $color(magenta)(cos(2theta))$ by $3$ $color(brown)3$ giving
$XXX 3 cos (2 θ)$ $color(white)("XXX")color(brown)(3cos(2theta))$

There is to be no horizontal shift, so the argument for $cos$ $cos$ will not be modified by any further addition/subtraction.

In order to achieve the vertical shift (that I assumed would be $+ 7$ $color(red)(+7)$ [substitute your own value]) we will need to add $7$ $color(red)7$ to all values in our modified range:
$XXX 3 cos (2 θ) + 7$ $color(white)("XXX")color(red)(3 cos(2theta) +7)$

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What cosine function represents an amplitude of 3, a period of π, no horizontal shift, and a vertical shift of?

1 Answer

Explanation:

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